Electrical power network modelling method

ABSTRACT

A versatile modelling for any standard power system component is disclosed, by which the component modules can be easily plugged into the network module to form a small perturbation state space model of the entire system irrespective of the complexity of the system. The state equation is available as explicitly a function of every parameter and the input and output can be any variable, thus providing insight into the physical nature by simple matrix manipulation.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to electrical power network modelling methods forsmall perturbation stability in a power transmission network.

2. Description of Prior Art

Power system stability has been and continues to be a major concern inthe system or network operation. Large and small disturbance are in twomain categories for stability analysis. In the past, the smalldisturbance performance was evaluated by the result of a transientstability program under a small perturbation. However, this simulationresult will provide a limited insight because of the difficulty intaking measurements and in ensuring sufficient stability margins forswing oscillations. The small disturbance analysis is increasinglyrecognized because the spontaneous nature of swing oscillations can beanalyzed based on a linearized system at the steady-state operatingpoint. An eigenvalue analysis of the system described in thisapplication can provide many insights which are difficult to be observedin transient plots.

Many methods have been proposed to represent networks, machines andassociated control equipment such as the excitation system (EXC),governor system (GOV) and power system stabilizer (PSS) as well as newcomponents of FACTs (flexible alternating current transmission) undersmall perturbation. In power system, the network and components aredescribed by equations, and the control equipments are usuallyrepresented by blocks. In all other existing techniques, the controlblocks are eventually transformed into equations in order to integratewith the network/component equations to form the state space equations.However, they have the following weakness:

(i) limited flexibility,

(ii) difficulty to interface with any user's new devices,

(iii) restricted input/output signal selection, e.g. ΔP_(m) and ΔV_(ref)for input signals,

(iv) infinite busbar assumption or restriction to a small hypotheticalsystem,

(v) difficulty for computer program implementation,

(vi) limited exploitation of eigenvector analysis.

SUMMARY OF THE INVENTION

According to the invention there is provided an electrical systemnetwork and component modelling method for small perturbation stabilityin a power system, the method comprising converting the system networkinto elementary transfer blocks, converting the components intoelementary transfer blocks, and plugging the components into the networkto form a state space model of the entire system.

The invention is distinguished from the existing techniques that,instead of converting block to equations, all equations (network andcomponent) are converted to blocks, such that blocks of controlequipment (to any degree of complexity) can be easily is amalgamated byassigning simple node numbers. Because the network equations is alsoconverted to blocks, any block modules of components, for instancemachines, static var compensator (SVC), phase shifter (PS), high voltagedirect current (HVDC) system and control and tieline, can be pluggedinto the network module.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described by way of examplewith reference to the accompanying drawings in which:

FIG. 1 is an overall view of a Plug-in Modelling Technique (PMT)connection;

FIG. 2 shows elementary transfer blocks;

FIG. 3 is a network representation with one machine and one othercomponent;

FIG. 4 is a third order machine module;

FIG. 5 is a fourth order machine module;

FIG. 6 is a fifth order machine module;

FIG. 7 is a sixth order machine module;

FIG. 8 is an SVC module;

FIG. 9 is a tieline module;

FIG. 10 is an example of a 4-block system;

FIG. 11 is an example of “non-elementary” transfer functions;

FIG. 12 is a formation of a ΔΩ_(COI)I;

FIG. 13 is a tieline infinite busbar network module;

FIG. 14 is a configuration of an SVC;

FIG. 15 is an SVC module;

FIG. 16 is a general SVC voltage control;

FIG. 17 is configuration of a TCSC;

FIG. 18 is a TCSC module;

FIG. 19 is a configuration of a phase shifter;

FIG. 20 is a PS module;

FIG. 21 is a configuration of an AC/DC network at one terminal;

FIG. 22 is an equivalent T-circuit of a HVDC link;

FIG. 23 is a DC transmission line module;

FIG. 24 is an HVDC link module;

FIG. 25 is a tie line module;

DESCRIPTION OF PREFERRED EMBODIMENTS

Embodiments of the invention provide pragmatic and highly versatilemethods, referred to as a Plug-in Modelling is Technique (PMT). Anystandard power system components such as a static var compensator (SVC)a thyristor controllable series compensator (TCSC), a phase shifter(PS), a high voltage direct current (HVDC) system and control, atieline, as well as a generator (of any degree of complexity) associatedwith control equipment can be modelled as modules and plugged into anetwork module (FIG. 1). Because all these system components can befully represented by two types of elementary transfer blocks and onlyfive types of parameters as shown in FIG. 2 (where “m” and “x” are thenon-state and state variables respectively), the state space equationcan be obtained easily by means of matrix manipulation irrespective ofthe complexity of the system. Stability of the system can be examinedbased on time/frequency response, eigenvalues, modal, and sensitivityanalysis.

In the network system modelling, the concept of the PMT is derived froma previous Generalized Multimachine Representation (GMR). In GMR,however, loads are considered as constant impedances and the systemcomponents are restricted to machines. In the present application PMT,the network formulation is much enhanced by the inclusion of loadcharacteristics and other system components. Moreover, options ofdifferent machine modelling are also provided.

For the stability studies, the load dynamics may be considered from:

S=P+jQ=P _(o)(V/V _(o))^(a) +jQ _(o)(V/V _(o))^(b)  (1)

Under small perturbation, the load characteristic can be represented byfour fictitious admittances in (2). $\begin{matrix}{{\begin{bmatrix}{\Delta \quad I_{R}} \\{\Delta \quad I_{J}}\end{bmatrix} = {\begin{bmatrix}Y_{RR} & Y_{RJ} \\Y_{JR} & Y_{JJ}\end{bmatrix}\begin{bmatrix}{\Delta \quad V_{R}} \\{\Delta \quad V_{J}}\end{bmatrix}}}{where}{Y_{RR} = {\frac{\left( {a - 2} \right){PV}_{R}^{2}}{V^{4}} + \frac{\left( {b - 2} \right){QV}_{R}V_{J}}{V^{4}} + \frac{P}{V^{2}}}}{Y_{RJ} = {\frac{\left( {a - 2} \right){PV}_{R}V_{J}}{V^{4}} + \frac{\left( {b - 2} \right){QV}_{J}^{2}}{V^{4}} + \frac{Q}{V^{2}}}}{Y_{JR} = {\frac{\left( {a - 2} \right){PV}_{R}V_{J}}{V^{4}} - \frac{\left( {b - 2} \right){QV}_{R}^{2}}{V^{4}} - \frac{Q}{V^{2}}}}{Y_{JJ} = {\frac{\left( {a - 2} \right){PV}_{J}^{2}}{V^{4}} - \frac{\left( {b - 2} \right){QV}_{R}V_{J}}{V^{4}} + \frac{P}{V^{2}}}}} & (2)\end{matrix}$

and subscripts RJ stand for the network frame.

Busbar injections, voltages and network admittance matrix are related byI=YV i.e. $\begin{matrix}{\begin{bmatrix}I_{G} \\{- I_{L}}\end{bmatrix} = {\begin{bmatrix}Y_{GG} & Y_{GL} \\Y_{LG} & Y_{LL}\end{bmatrix}\begin{bmatrix}{\quad V_{G}} \\{\quad V_{L}}\end{bmatrix}}} & (3)\end{matrix}$

where subscripts G and L stand for generators and loads. Applying smallperturbation and splitting the complex Y_(NxN) matrix of (3) into realY_(2Nx2N) matrix (N=number of nodes), the load current I_(L) iseliminated by equivalents given by (2), which is then included the blockdiagonal elements of Y, to obtain $\begin{matrix}{\begin{bmatrix}{\Delta \quad I_{G}} \\0\end{bmatrix} = {\begin{bmatrix}Y_{GG} & Y_{GL} \\Y_{LG} & Y_{LL}\end{bmatrix}\begin{bmatrix}{\Delta \quad V_{G}} \\{\Delta \quad V_{L}}\end{bmatrix}}} & (4)\end{matrix}$

Thus the load buses can be eliminated to $\begin{matrix}{{\left\lbrack {\Delta \quad I_{G}} \right\rbrack = {\left\lbrack Y_{GG}^{\prime} \right\rbrack \quad\left\lbrack {\Delta \quad V_{G}} \right\rbrack}}{{{where}\quad Y_{GG}^{\prime}} = {Y_{GG} - {{Y_{GL}\left( Y_{LL}^{\prime} \right)}^{- 1}Y_{LG}}}}} & (5)\end{matrix}$

where Y_(GG)′=Y_(GG)−Y_(GL)(Y_(LL)′)⁻¹Y_(LG)

Subsequently, only the generator buses are retained. However, ifconnected buses associated with other system Components are alsoconsidered, the network equation can be extended to $\begin{matrix}{\begin{bmatrix}{\Delta \quad I_{G}} \\{{- \Delta}\quad I_{C}}\end{bmatrix} = {\begin{bmatrix}Y_{GG} & Y_{GC} \\Y_{CG} & Y_{CC}\end{bmatrix}\begin{bmatrix}{\Delta \quad V_{G}} \\{\Delta \quad V_{C}}\end{bmatrix}}} & (6)\end{matrix}$

where ΔI_(c) is the component current, ΔV_(c) is the component busbarvoltage and subscripts G and C stand for generator and other systemcomponents. Whilst generator terminals must be in series with a uniquegenerator transformer, a component busbar can have multi-branch currentflow. Therefore, the roles of the ΔV_(c) and ΔI_(c) in the networkequation (6) must be swapped such that all components connected to thesame busbar have a common input ΔV_(c). The modified network equationbecomes $\begin{matrix}{\begin{bmatrix}{\Delta \quad I_{{RJ}_{G}}} \\{\Delta \quad I_{{RJ}_{C}}}\end{bmatrix} = {\begin{bmatrix}Y_{GG} & C_{GC} \\C_{CG} & Z_{CC}\end{bmatrix}\begin{bmatrix}{\Delta \quad V_{{RJ}_{G}}} \\{\Delta \quad I_{{RJ}_{C}}}\end{bmatrix}}} & (7)\end{matrix}$

where Y_(GG)′=Y_(GC)Y_(CC) ⁻¹Y_(CG), C_(GC)=Y_(CC) ⁻¹, C_(CG)=−Y_(CC)⁻¹Y_(CG) and Z_(CC)=Y_(CC) ⁻¹. Note that LHS (Left Hand Side) of (7),the “output” of network, is the “input” of components.

For instance, if the system consists one machine and one other shuntcomponent, the full matrix equation will be $\begin{matrix}{\begin{bmatrix}\begin{matrix}\begin{matrix}{\Delta \quad I_{R1}} \\{\Delta \quad I_{J1}}\end{matrix} \\\overset{\underset{\quad}{\_}}{\Delta \quad V_{R2}}\end{matrix} \\{\Delta \quad V_{J2}}\end{bmatrix} = {\begin{bmatrix}{Y11} & {Y12} & {C13} & {C14} \\{Y21} & {Y22} & {C23} & {C24} \\{C31} & {C32} & {Z33} & {Z34} \\{C41} & {C42} & {Z43} & {Z44}\end{bmatrix}\begin{bmatrix}\begin{matrix}\begin{matrix}{\Delta \quad V_{R1}} \\{\Delta \quad V_{J1}}\end{matrix} \\\overset{\underset{\quad}{\_}}{\Delta \quad I_{R2}}\end{matrix} \\{\Delta \quad I_{J2}}\end{bmatrix}}} & (8)\end{matrix}$

and these two components are connected by the network transfer blocks asshown in FIG. 3.

In creating the representations of the network components, a machine isderived from Park's two-axis machine model. Machine behaviours can bedescribed by different sets of equations of different order in themachine frame.

For instance, the 3rd-order machine equations in Park's dq-frame are

pE _(q) ′=[E _(jd)−(X _(d) −X _(d)′)I _(d) −E _(q) ′]/T _(do)′  (9a)

pδ/Ω ₀=Ω−1  (9b)

pΩ=[P _(m) −P _(c) ]/M  (9c)

V _(f) ² =V _(d) ² +V _(q) ²  (9d)

V _(d) =−R _(a) I _(d) +X _(q) I _(q)  (9e)

V _(q) =E _(q) ′−X _(d) ′I _(d) −R _(a) I _(q)  (9f)

P _(c) =V _(q) I _(q) +V _(d) I _(d)+(I _(d) ² +I _(q) ²)R _(a)  (9g)

By applying small perturbation to the above equations:

ΔE _(q) ′=[ΔE _(jd)−(X _(d) −X _(d)′)ΔI _(d)]/(1+pT _(do)′)  (10a)

Δδ=Ω₀ ΔΩ/p  (10b)

ΔΩ=(ΔP _(m) −ΔP _(c))/pM  (10c)

ΔV _(r) =V _(d) ΔV _(d) /V _(f) +V _(q) ΔV _(q) /V _(r)  (10d)

ΔV _(d) =−R _(a) ΔI _(d) +X _(q) ΔI _(q)  (10e)

ΔV _(q) =ΔE _(q) ′−X _(d) ′ΔI _(d) −R _(o) ΔI _(q)  (10f)

ΔP _(c)=(V _(q)+2R _(a) I _(q))ΔI _(q) +I _(q) ΔV _(q)+(V _(d)+2R _(a) I_(d))ΔI _(d) +I _(d) ΔV _(d)  (10g)

Equations (10) can be expressed by blocks or modules together with theblock model of control equipment such EXC, GOV and PSS (FIG. 4). Thus, aself-contained individual machine module is formed as shown in FIG. 4.Other machine models can be similarly developed as shown in FIGS. 5, 6and 7.

For a machine and network interface, the generator will have its own dqframe which can be related to the network (RJ) frame by

I _(d) =I _(R) COS δ+I _(J) sin δ  (11a)

I _(q) =−I _(R) sin δ+I _(J) COS δ  (11b)

V _(R) =V _(d) COS δ−V _(q) sin δ  (11c)

V _(J) =V _(d) sin δ+V _(q) COS δ  (11d)

Under small perturbation, take the form:

 ΔI _(d)=COS δΔI _(R)+sin δΔI _(J) ÷I _(q)Δδ  (12a)

ΔI _(q)=−sin δΔI _(R)+COS δΔI _(J) −I _(d)Δδ  (12b)

ΔI′ _(R)=COS δΔI ⁻ _(d)−sin δΔV _(q) −V _(J)Δδ  (12c)

ΔV _(J)=sin δΔV _(d)+COS δΔV _(q) ÷V _(R)Δδ  (12d)

where δ is the angle between the dq frame and RJ frame.

Thus, the machine model can be “plugged” into the network by an adapteras shown in FIG. 3.

Formation of other component modules (such as SVC, TCSC, PS, HVDC andtieline are provided in TFDEPC. All these modules, with inputs (ΔV_(R),ΔV_(J)) and outputs (ΔI_(R), ΔI_(J)), can therefore be “plugged” intothe network module (FIG. 1).

In network or system formation and analysis, using PMT the blockstructure is so general, computer algorithms (dealing only with twotypes of blocks and five parameters) can be easily programmed without amatrix format being concerned.

To run the program, a user has to

(a) prepare and perform a loadflow analysis

(b) input component data of the system (taking all the blocks, withtheir node numbers, within FIG. 1)

(c) input block data of the controller for the system components such asEXC, GOV and PSS (the “standard” models may also be retrieved fromseparate data files if preferred)

(d) start computation

The procedure of deriving the state equations (13) and (14), see below,or evaluating the sensitivity coefficients is unaffected by thecomplexity of the physical system is shown in TFDEPC 9.2 and TFDEPC 9.6.Because the entire system consists of only elementary blocks, existingsoftware such as Matlab can be used as an analytical tool and advancedcontrol techniques provided by the toolbox of Matlab can be applieddirectly and easily.

{dot over (X)}=AX+BR+E{dot over (R)}  (13)

Y=CX+DR  (14)

where the E{dot over (R)} term is often avoided in the usualrepresentation but it does not create any analytical difficulty in thestability study.

Previously, the equipment representation (such as EXC) is limited tosome specified models in order to save programming time for some“standard” packages. Due to the large variety of manufactured systems inpractical use, simplification of this actual model to a specific modelmay provide a misleading result. It is very tedious to adjust theoriginal EXC blocks (based on data easily obtained from themanufacturer's design) to fit them into some unrelated “standard”models. It would also be difficult to determine the optimum settings ofspecific control cards (e.g. the lead/lag time constants in the EXCfeedback loop) since the “standard” model may contain no such blocks.Moreover, employing the “standard” model may lead to an improper PSSdesign, because the optimum setting is quite dependent on the EXC-PSSfrequency response. The PMT of this patent application, however,facilitates the modelling of machines system components and theircontrol equipment to any degree of complexity, including those IEEEmodels described by nonlinear functions or quadratic/polynomial transferfunctions involving complex poles and zeros, as they can be convertedreadily to the elementary blocks of FIG. 2 according to TFDEPC 9.4.

Due to the improvement in solid state technology, a lot of newcomponents are introduced such as FACTS. In order to include these kindsof components, users need to form the plant using his own programbecause it is very difficult (if not impossible) for a general user tointegrate these new components into the existing modelling methods. Dueto the modular structure of the PMT, any new system component modulescan be “plugged” in the network model via the specific input/outputvariables described in FIG. 1. So there is no restriction to investigatesystem stability by consideration of the interaction among anycomponents and the solid-state development of the power system networksapplications can be enhanced.

In controller designs, the output vectors are restricted to statevariables, e.g. speed or their linear combinations. This is probablybecause of the difficulty of deriving (14) for some non-state variableoutputs e.g. ΔP_(e). In the so-called PQR techniques, the outputvariables are predefined to describe the network equation, whether theyare used in subsequent system analysis or not. Due to this specified andlimited input/output signals, it is difficult to take any unusualinput/output signal for some typical controller design. In the PMTarrangement, any locally available variables such as voltage, current,MW, MVAr or MVA etc is can be used as control signals.

In the choice of reference frame, the infinite busbar, first introducedto reduce a system to a single machine equivalent but subsequently usedto study multimachine systems, does not exist in reality. Instead ofrelying on an infinite busbar, an alternative assumption is that“network frequency” corresponds always to that of one arbitrarily chosenmachine. However, in PMT, any machine speed can be used as a referenceby means of a simple “−1” block (i.e. the connection of ΔΩ_(ref) in FIG.4). A more reasonable or practical reference frame is to use the centerof inertia (COI) of the entire network (or within a specific area) asshown in TFDEPC 9.5. Moreover, this COI frequency can be also used as adamping signal as described above.

Although eigenvalue technique has long been recognised as an effectivetool to analyse power system small perturbation stability, the potentialof eigenvalue analytical tools (modal and sensitivity analyses) has sofar not been fully realized in other stability studies. The mainrestriction has been associated with the way in which the system stateequations are normally formulated. The advantages of PMT method arebrought out by the relative ease of its integration with the followingtechniques for practical synthesis.

For modal analysis in the PMT method, mode shape for any variableassociated with the eigenvalue λ=α+jω can be determined easily from CU,where U is the eigenvector and C is the coefficient matrix in (14).Modal analysis for non-state variables was not applied. This may beattributed to the difficulty in establishing C if the y's in Y of (14)have to contain arbitrary non-state variables, for example, ΔP_(e),ΔI_(d) or ΔI_(q). This is equivalent to expressing ΔP_(e), say, in termsof all the state variables (x's) of the machines, including EXC, PSS orgovernor, as well as of other components. In the PMT method, however,the y's can be chosen as any system variables or their combinationwithout any restriction For sensitivity for arbitrary block parametersin the PMT method, matrix A is a function of all transfer blockparameters so that the final expressions of ∂λ/∂κ are only simplealgebraic scalar operations (e.g. ∂λ/∂b=wz/(hT_(a))) where the scalars(e.g. w,z and h) can be derived very easily from U and V as shown inTFDEPC 6.6.1. (“T_(a)” and “b” are known transfer block parameters.)

The sensitivity expression for a change in block parameter is simple andstraight forward. However, if the change is on a system parameter,numerous block parameters will be affected. For instance, the change ina bus MW loading P_(Li) will affect all nodal voltages and the operatingconditions of generators, which results in the change of some machineblock parameters. Change in a line reactance X_(ij) will furthermoreaffect the block parameters associated with the elements of matricesY_(GC)′, C_(GC), C_(CG) and Z_(CC) in the network equation (7).Fortunately, these changes, although very numerous, only affect them-block parameters. With properly reordering the m-blocks as:

M=[M₁, M₂/M₃]  (16)

where

M₁=parameters varying with nodal voltages,

M₂=parameters being the elements of matrices Y_(GG)′, C_(GC), C_(CG) andZ_(CC),

M₃=the rest constant parameters

it is still possible to obtain ∂A/∂κ from the partial derivatives of them-block parameters. Matrices Y_(GC)′, C_(GC), C_(CG) and Z_(CC)associated with M₂ are created from the blocked matrix manipulation ofnodal admittance matrix Y_(GG), Y_(GC), Y_(CG) and Y_(CC) in (6). Whilethe block parameters belonging to M₁, such as V_(d), V_(q), I_(d),I_(q), V_(t), cosδ and sinδ etc., can be expressed by busbar voltages (Vand δ), even though these buses may have been eliminated in (7). Finallyit is possible to evaluate ∂λ/∂κ, for κ=P_(j), Q_(j) (arbitrary businjection), G_(ij) or B_(ij) (arbitrary circuit admittance), because thedifferentiation of V or δ with respect to nodal injections has beendetermined in the loadflow algorithm.

The complicated sensitivity expression for system parameters aresummarized in TFDEPC 9.6.2 for constant admittance load and third ordermachine model only. These equations can be directly applied to ahigher-order machine model, or easily extended to other systemcomponents and load representations. What is needed is only thereordering of m-blocks and the reconstruction of M₁, M₂ and M₃. Thesensitivity technique ∂λ/∂κ for κ=V_(R), V_(J) has been applied to theprobabilistic analysis of load variation , the sensitivity ∂λ/∂κ forκ=Q_(L) has been applied to identify the best SVC location. The secondorder sensitivities ∂²λ/(∂κ_(i)∂κ_(j)) for κ_(i)=V_(R), V_(J) andK_(j)=(PSS gain) can be used for evaluating the probabilisticdistribution of sensitivity coefficient.

The validation of the PMT algorithms can be divided into two stages. Thefirst stage (in three steps) is to compare the results with thewell-known Heffron Phillips (HP) model.

(a) Machine modelling is first checked by using the new single machineinfinite bus model as established in TFDEPC 9.8.

(b) The multimachine network described by (5) is checked against HPusing a 2-machine 2-node system in which one machine (reference) hasvery large inertia and the output voltages ΔV_(d) and ΔV_(q) are madezero.

(c) This network equation (5) is once more checked by a large N-machinesystem in which two identical machines are connected to an arbitrary busvia identical reactance X_(e). (e.g. generator transformer reactance)

The above will confirm the validity of the 3rd-order machine modeltogether with the network equation (5) because all the eigenvaluesobtained in HP analysis reappear in all the above cases irrespective thecomplexity of EXC/GOV representations, although some more eigenvaluesare present in (b) and (c) because there is more than one machine. Thenext stage is to check the network equation (7) (with any component)with the validated network equation (5) (with machines only):

(d) by treating an arbitrary transmission line (eventually eliminated in(5)) as a line component in (7),

(e) by treating a capacitor as a SVC component without feedback control,

(f) with the existence of both multi-SVC and multi-line components.

The validity of the network equation (7) can also be confirmed becauseeigenvalues obtained in computer runs are identical to the original(i.e. without SVC/line component). Note that the shunt SVC componentconnected to one node (FIG. 8) and the series line component connectedbetween two nodes (FIG. 9) reflect the different ways of modular systemconnection.

Transfer Function Descriptions of Electric Power Components (Referred inthe Specification as “TFDEPC”)

9.1.2.1 Static Var Compensator (SVC)

A typical SVC configuration with thyristor controlled reactor & fixedcapacitor connected to a busbar through a step-down transformer is shownin FIG. 14 where X_(r) is the transformer impedance, B_(C) and B_(L) arethe susceptance of capacitor and inductor. The SVC equations can bewritten as $\begin{matrix}{X_{SVC} = {X_{T} - \frac{1}{B_{C} + B_{L}}}} & (108) \\{\begin{bmatrix}V_{R} \\V_{J}\end{bmatrix} = {X_{SVC}\begin{bmatrix}{- I_{J}} \\I_{R}\end{bmatrix}}} & (109)\end{matrix}$

Substituting eqn.108 in eqn.109 and applying small perturbation, eqn.110 can be obtained and the block diagram of SVC is created in FIG. 15.Besides, the general SVC thyristor controller block diagram is includedin FIG. 16 where V_(SVC) and I_(SVC) are the magnitude of voltage andcurrent at the SVC terminal [16]. $\begin{matrix}{\begin{bmatrix}{\Delta \quad I_{R}} \\{\Delta \quad I_{J}}\end{bmatrix} = {\frac{1}{X_{SVC}}\left\{ {\begin{bmatrix}{\Delta \quad V_{J}} \\{{- \Delta}\quad V_{R}}\end{bmatrix} - {{\frac{1}{\left( {B_{C} + B_{L}} \right)^{2}}\begin{bmatrix}I_{R} \\I_{J}\end{bmatrix}}\Delta \quad B_{L}}} \right\}}} & (110)\end{matrix}$

9.1.2.3 Thyristor Controlled Series Compensator (TCSC)

A typical TCSC configuration with thyristor controlled reactor & fixedcapacitor connected in series with the transmission line is shown inFIG. 17 and described by $\begin{matrix}{X_{TCSC} = {- \frac{1}{B_{C} + B_{L}}}} & (111) \\{\begin{bmatrix}{V_{R1} - V_{R2}} \\{V_{J1} - V_{J2}}\end{bmatrix} = {X_{TCSC}\begin{bmatrix}{- I_{J1}} \\I_{R1}\end{bmatrix}}} & (112) \\{\begin{bmatrix}I_{R1} \\I_{J1}\end{bmatrix} = {- \begin{bmatrix}I_{R2} \\I_{J2}\end{bmatrix}}} & (113)\end{matrix}$

By applying small perturbation to the above equations and rearranging,eqn.114 can be obtained and the block diagram for the TCSC can berepresented by FIG. 18. $\begin{matrix}{\begin{bmatrix}{\Delta \quad I_{R1}} \\{\Delta \quad I_{J1}}\end{bmatrix} = {\frac{1}{X_{TCSC}}\left\{ {\begin{bmatrix}{{\Delta \quad V_{J1}} - {\Delta \quad V_{J2}}} \\{{{- \Delta}\quad V_{R1}} + {\Delta \quad V_{R2}}}\end{bmatrix} - {{\frac{1}{\left( {B_{C} + B_{L}} \right)^{2}}\begin{bmatrix}I_{R1} \\I_{J1}\end{bmatrix}}\Delta \quad B_{L}}} \right\}}} & (114)\end{matrix}$

9.1.2.4 Phase Shifter (PS)

Consider a phase shifter connected between busbars 1 & 2 and representedby voltage ratio T=1∠ψ in series with transformer admittance (y) asshown in FIG. 18, eqn.115 would be obtained. $\begin{matrix}{\begin{bmatrix}I_{1} \\I_{2}\end{bmatrix} = {{y\begin{bmatrix}1 & {- T} \\{- T^{*}} & 1\end{bmatrix}}\begin{bmatrix}V_{1} \\V_{2}\end{bmatrix}}} & (115)\end{matrix}$

By applying small perturbation to eqn.115, then $\begin{matrix}{\begin{bmatrix}{\Delta \quad I_{1}} \\{\Delta \quad I_{2}}\end{bmatrix} = {{{y\begin{bmatrix}1 & {- T} \\{- T^{*}} & 1\end{bmatrix}}\begin{bmatrix}{\Delta \quad V_{1}} \\{\Delta \quad V_{2}}\end{bmatrix}} + {{y\begin{bmatrix}{{- j}\quad V_{2}T} \\{j\quad V_{1}T^{*}}\end{bmatrix}}\Delta \quad \psi}}} & (116)\end{matrix}$

Splitting the complex matrix to real variable matrix, eqn.117 and FIG.19 can be obtained. $\begin{matrix}{\begin{bmatrix}\begin{matrix}\begin{matrix}{\Delta \quad I_{R1}} \\{\Delta \quad I_{J1}}\end{matrix} \\{\Delta \quad I_{R2}}\end{matrix} \\{\Delta \quad I_{J2}}\end{bmatrix} = {{\begin{bmatrix}G_{11} & {- B_{11}} & G_{12} & {- B_{11}} \\B_{11} & G_{11} & B_{12} & G_{12} \\G_{21} & {- B_{21}} & G_{22} & {- B_{22}} \\B_{21} & G_{21} & B_{22} & G_{22}\end{bmatrix}\begin{bmatrix}\begin{matrix}\begin{matrix}{\Delta \quad V_{R1}} \\{\Delta \quad V_{J1}}\end{matrix} \\{\Delta \quad V_{R2}}\end{matrix} \\{\Delta \quad V_{J2}}\end{bmatrix}} + {\begin{bmatrix}\begin{matrix}\begin{matrix}K_{R1} \\K_{J1}\end{matrix} \\K_{R2}\end{matrix} \\K_{J2}\end{bmatrix}{\Delta\psi}}}} & (117)\end{matrix}$

i.e.

[ΔI _(RJ) ]=[Y _(PS) ][ΔV _(RJ) ]+[K _(PS)]Δψ  (118)

9.1.2.5 High Voltage Direct Current System

Configuration of AC/DC network at each terminal is shown in FIG. 20.

Converter Equation (117)

V _(DC) =V _(do) COS α−R _(C) I _(DC)  (119)

V _(R) I _(R) +V _(J) I _(J) =V _(DC) I _(Dc)  (120)

I ² +I _(J) ² =k _(DC) I _(DC) ²  (121)

where $k_{DC} = {\frac{18}{\pi^{2}}a^{2}k^{2}n^{2}B^{2}}$

V_(do)=3(2)αV_(t)B/π

and

is the communication overlap factor,

n is the number of poles per terminal,

B is the number of bridges in series for each pole.

a is the off-nominal turns ration of the transformer

Applying small perturbation, $\begin{matrix}{{\Delta \quad V_{DC}} = {{\frac{3\sqrt{2}}{\pi}{aB}\quad \cos \quad \alpha \quad \Delta \quad V_{t}} + {V_{do}\Delta \quad \cos \quad \alpha} - {R_{C}\Delta \quad I_{DC}}}} & (122) \\{\begin{bmatrix}{\Delta \quad I_{R}} \\{\Delta \quad I_{J}}\end{bmatrix} = {{R\begin{bmatrix}{\Delta \quad I_{DC}} \\{\Delta \quad V_{DC}}\end{bmatrix}} + {N\begin{bmatrix}{\Delta \quad V_{R}} \\{\Delta \quad V_{J}}\end{bmatrix}}}} & (123)\end{matrix}$

where $R = {{\frac{1}{M}\begin{bmatrix}{{k_{DC}I_{DC}V_{J}} - {{nI}_{J}I_{DC}}} & {{- {nI}_{J}}I_{DC}} \\{{{- k_{DC}}I_{DC}V_{R}} + {{nI}_{R}V_{DC}}} & {{nI}_{R}I_{DC}}\end{bmatrix}} = \begin{bmatrix}R_{11} & R_{12} \\R_{21} & R_{22}\end{bmatrix}}$ $N = {{\frac{1}{M}\begin{bmatrix}{I_{R}I_{J}} & I_{J}^{2} \\{- I_{R}^{2}} & {{- I_{R}}I_{J}}\end{bmatrix}} = \begin{bmatrix}N_{11} & N_{12} \\N_{21} & N_{22}\end{bmatrix}}$

and

M=I_(R)V_(J)−I_(J)V_(R)

Current Regulator Equation $\begin{matrix}{{p\quad \cos \quad \alpha} = {{\frac{K_{C}}{T_{C}}\left( {I_{DC} - I_{ref} - U_{C}} \right)} - {\frac{1}{T_{C}}\cos \quad \alpha}}} & (124)\end{matrix}$

Under small perturbation, $\begin{matrix}{{\Delta \quad \cos \quad \alpha} = {\frac{K_{C}}{1 + {p\quad T_{C}}}\left( {{\Delta \quad I_{DC}} - {\Delta \quad I_{ref}} - {\Delta \quad U_{C}}} \right)}} & (125)\end{matrix}$

i.e. The firing angle (α) in eqns.119,122,124 and 125 will become theadvance angle (β) and I_(DC) will become negative for inverter side.

DCA Network

If a T-model is used to represent the DC line (FIG. 22), then

V _(DC1) =R ₁ I _(DC1) +L ₁ pI _(DC1) +V _(C)  (126)

V _(DC2) =R ₂ I _(DC2) +L ₂ pI _(DC2) +V _(C)  (127)

$\begin{matrix}{V_{C} = {\frac{1}{Cp}\left( {I_{DC1} + I_{DC2}} \right)}} & (128)\end{matrix}$

Hence $\begin{matrix}{{\Delta \quad I_{DC1}} = \frac{{\Delta \quad V_{DC1}} - {\Delta \quad V_{C}}}{R_{1} + {L_{1}p}}} & (129) \\{{\Delta \quad I_{DC2}} = \frac{{\Delta \quad V_{DC2}} - {\Delta \quad V_{C}}}{R_{2} + {L_{2}p}}} & (130) \\{{\Delta \quad V_{C}} = {\frac{1}{Cp}\left( {{\Delta \quad I_{DC1}} + {\Delta \quad I_{DC2}}} \right)}} & (131)\end{matrix}$

Therefore, the block diagram of a DC transmission line is formed as FIG.22 and the HVDC link module can be obtained as shown in FIG. 23.

9.1.2.6 Tieline

Tieline flow is one of the very effective damping signal. In order toget/monitor this signal, the tieline (or any circuit) has to be treatedas one of the system components. Since the terminal voltages & currentsof a tieline can be related by: $\begin{matrix}{\begin{bmatrix}I_{1} \\I_{2}\end{bmatrix} = {\begin{bmatrix}Y_{11} & Y_{12} \\Y_{21} & Y_{22}\end{bmatrix}\begin{bmatrix}V_{1} \\V_{2}\end{bmatrix}}} & (132) \\{\begin{bmatrix}{\Delta \quad I_{1}} \\{\Delta \quad I_{2}}\end{bmatrix} = {\begin{bmatrix}Y_{11} & Y_{12} \\Y_{21} & Y_{22}\end{bmatrix}\begin{bmatrix}{\Delta \quad V_{1}} \\{\Delta \quad V_{2}}\end{bmatrix}}} & (133)\end{matrix}$

By splitting the complex matrix to real variable matrix,

[ΔI_(RJ)]=[Y_(TL)][ΔV_(RJ)]  (134)

and the tieline model is shown in FIG. 9. Thus any tieline signals ateither side can be obtained by $\begin{matrix}{{\Delta \quad P} = {{V_{R}\Delta \quad I_{R}} + {I_{R}\Delta \quad V_{R}} + {V_{J}\Delta \quad I_{J}} + {I_{J}\Delta \quad V_{J}}}} & (135) \\{{\Delta \quad Q} = {{V_{J}\Delta \quad I_{R}} - {I_{J}\Delta \quad V_{R}} - {V_{R}\Delta \quad I_{J}} + {I_{R}\Delta \quad V_{J}}}} & (136) \\{{\Delta \quad S} = {{\frac{P}{S}\Delta \quad P} + {\frac{Q}{S}\Delta \quad Q}}} & (137) \\{{\Delta \quad I} = {{\frac{I_{R}}{I}\Delta \quad I_{R\quad}} + {\frac{I_{J}}{I}\Delta \quad I_{J}}}} & (138) \\{{\Delta \quad V} = {{\frac{V_{R}}{V}\Delta \quad V_{R\quad}} + {\frac{V_{J}}{V}\Delta \quad V_{J}}}} & (139)\end{matrix}$

9.2 Formation of State Space Equation

In the PMT, the blocks are either of zero order, as in FIG. 2a, or offirst order as in FIG. 2b. The “in” and “out” of each block are assignedwith node numbers. An example of connection of a four-block system isshown in FIG. 12 and referred to in TFDEPC 9.3. Thus, referring to therespective node numbers, the connection L-matrix can be constructed inthe following form: $\begin{matrix}{\begin{bmatrix}\begin{matrix}X_{i} \\Y\end{matrix} \\M_{i}\end{bmatrix} = {\begin{bmatrix}L_{1} & L_{2} & L_{3} \\L_{4} & L_{5} & L_{6} \\L_{7}^{\prime} & L_{8}^{\prime} & L_{9}^{\prime}\end{bmatrix}\begin{bmatrix}X \\R \\M\end{bmatrix}}} & \text{(17a)(17b)(17c)}\end{matrix}$

where X_(i), X, M_(i) and M are vectors collecting all the x_(i), x,m_(i) and m variables of FIG. 2, and R and Y are the input and outputvectors. As the equation of the nth zero order block ism_(n)=k_(n)m_(i), the matrix equation for all zero order blocks will be

M=KM_(i)  (18)

where

K=<k _(n)>  (19)

=diagonal matrix of k_(n), n=1,2, . . . .

Similarly, the equation of the first order block is

x _(n) =x _(i) _(n) (b _(n) +pT _(b) _(n) )/(a _(n) +pT _(a) _(n))  (20)

Expanding (20), the equation for all the first order blocks is

 X=−K _(c) X+K _(b) X _(i) +K _(f) {dot over (X)} _(i)  (21)

where

K _(a) =<k _(a) _(n) >=<a _(n) /T _(a) _(n) >  (22a)

K _(b) =<k _(b) _(n) >=<b _(n) /T _(a) _(n) >  (22b)

and

K _(f) =<k _(r) _(n) >=<T _(b) _(n) /T _(a) _(n) >  (22c)

are diagonal matrices (n=1,2 . . . ).

The state space equation can be constructed from (17), (18) and (21) byeliminating M_(i), M and then X_(i) from (17).

Therefore, substituting (17c) in (18),

M=KM _(i) =KL ₇ ′X+KL ₈ ′R+KL ₉ M=L ₇ X+L ₈ R+L ₉ M

so

M=H(L ₇ X+L ₈ R)  (23)

where

H=(I−L ₉)⁻¹

and

I=identity matrix

Substituting (23) in (17b),

Y=L ₁ X+L ₅ R+L ₆ H(L ₇ X+L ₈ R)

so

Y=CX+DR  (14)

where

C=L ₁ +L ₆ HL ₇

and

D=L ₅ +L ₆ HL ₈

Substituting (23) in (17a),

X _(i) =L ₁ X+L ₂ R+L ₃ H(L ₇ X+L ₈ R)

so

X _(i) =FX+GR  (24)

where

F=L ₁ +L ₃ HL ₇  (25)

and

G=L ₂ +L ₃ HL ₈

Substituting (24) in (21),

{dot over (X)}=−K _(a) X+K _(b)(FX+GR)÷K _(i)(F{dot over (X)}+G{dot over(R)})

so that

{dot over (X)}=AX+BR+E{dot over (R)}  (13)

where

A=S(K _(b) F−K _(a))  (26)

B=SK_(b)G

E=SK_(c)G

and

S=(I−K _(c) F)⁻¹  (27)

9.3 Formation of the L-Matrix

With reference to the simple example of a four-block system shown inFIG. 12, and by comparing the corresponding node numbers of the LHS andRHS vectors (“1” if equal otherwise “0”), the sparse connection-matrixequation is ${\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}(12) \\(13)\end{matrix} \\(13)\end{matrix} \\(11)\end{matrix} \\(14)\end{matrix}\begin{bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}x_{i_{1}} \\x_{i_{2}}\end{matrix} \\\overset{\_}{\underset{\_}{\overset{\quad}{y}}}\end{matrix} \\m_{i_{1}}\end{matrix} \\m_{i_{2}}\end{bmatrix}} = {{\left\lbrack \overset{\begin{matrix}{(13)} & {(14)} & {\quad {(11)}} & {(12)} & {(11)}\end{matrix}}{\begin{matrix}\quad & \quad & \quad & 1 & \quad \\{1\quad} & \quad & \quad & \quad & \quad \\{1\quad} & \quad & \quad & \quad & \quad \\\quad & \quad & {\quad 1\quad} & \quad & {\quad 1} \\\quad & {1\quad} & \quad & \quad & \quad\end{matrix}} \right\rbrack \begin{bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}x_{1} \\x_{2}\end{matrix} \\\overset{\_}{\underset{\_}{\overset{\quad}{r}}}\end{matrix} \\m_{1}\end{matrix} \\m_{2}\end{bmatrix}}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}(13) \\(14)\end{matrix} \\(11)\end{matrix} \\(12)\end{matrix} \\(11)\end{matrix}}$

Since the input data are a sequence of blocks (i.e. branch) rather thannodal information, this technique is much simpler and faster than theindirect method of using incidence matrices in this case.

9.4 Representation of Complicated Transfer Functions

Nonlinear function under small perturbations, or “quadratic” transferfunctions, may be expressed as elementary blocks in FIG. 2, asillustrated for some typical cases shown in FIG. 13.$\text{Case a:}\quad \begin{matrix}{Z = \quad {X\quad Y}} \\{{\Delta \quad Z} = \quad {{Y\quad \Delta \quad X} + {X\quad \Delta \quad Y}}}\end{matrix}$ $\text{Case b:}\quad \begin{matrix}{E = \quad {{V + {j\quad I\quad X}}}} \\{E^{2} = \quad {V^{2} + {I^{2}X^{2}}}} \\{{\Delta \quad E} = \quad {{\left( {V/E} \right)\Delta \quad V} + {\left( {X^{2}{I/E}} \right)\Delta \quad I}}}\end{matrix}$ $\text{Similarly}\quad \begin{matrix}{I = \quad {{I_{d} + {j\quad I_{q}}}}} \\{{\Delta \quad I} = \quad {{\left( {I_{d}/I} \right)\Delta \quad I_{d}} + {\left( {I_{q}/I} \right)\Delta \quad I_{q}}}}\end{matrix}$ $\text{Case c:}\quad \begin{matrix}{E = \quad {f(I)}} \\{{\Delta \quad E} = \quad {m\quad \Delta \quad I}}\end{matrix}$where  m = slope  (∂E/∂I)  at  the  operating  point.Case d:    A ’quadratic’ transfer function  may berepresented by two elementary blocks.  The  two blockdiagrams are identical so long as$\begin{matrix}{\quad {G = \quad C_{1}}} \\{A = \quad {D_{0}{C_{1}/C_{0}}}} \\{K = \quad {{C_{0}/C_{1}^{2}} + {D_{0}/C_{0}} - {D_{1}/C_{1}}}} \\{B = \quad {C_{0}/C_{1}}}\end{matrix}$

In the limiting case where C_(o)=0, C₁=0, or if the numerator polynomialis also quadratic, it is still possible to express the original secondorder block as a combination of elementary blocks by more elaboratecircuit block manipulation. Being a combination of quadratic and firstorder, the high order polynomial can be also represented by elementaryblocks.

9.5 COI Reference Frame

The position of the COI is defined as $\begin{matrix}{\delta_{COI} = {\sum\limits_{i = 1}^{p}{k_{i}\delta_{i}}}} & (28)\end{matrix}$

where k_(i)=H_(i)/H_(T) and H_(i), H_(T) are the individual and the sumof the inertia constant of the generator. δ_(i) is the position ofgenerator i with respect to the stationary frame. $\begin{matrix}{{i.e.\quad \Omega_{COI}} = {\sum\limits_{i = 1}^{p}{k_{i}\Omega_{i}}}} & (29)\end{matrix}$

Applying the small perturbation to (29), (30) and the COI speed signalare obtained (FIG. 14). $\begin{matrix}{{\Delta \quad \Omega_{COI}} = {\sum\limits_{i = 1}^{p}{k_{i}\Delta \quad \Omega_{i}}}} & (30)\end{matrix}$

9.6 Evaluation of Sensitivity Coefficients ∂λ/∂κ

9.6.1. Sensitivity for Arbitrary Block Parameters

From (26) and (27), system matrix A is a function of K_(b), K_(a) andK_(t). Hence, for κ=k_(b), k_(a) or k_(t) in the nth block as defined in(22), ∂A/∂κ can be expressed as follows: $\begin{matrix}{\frac{\partial A}{\partial k_{b}} = {S\frac{\partial K_{b}}{\partial k_{b}}F}} & \text{(31a)} \\{{\frac{\partial A}{\partial k_{a}} = {{- S}\frac{\partial K_{a}}{\partial k_{a}}}}{or}} & \text{(31b)} \\\begin{matrix}{\frac{\partial A}{\partial k_{t}} = \quad {\frac{\partial S}{\partial k_{t}}\left( {{K_{b}F} - K_{a}} \right)}} \\{= \quad {{- S}\frac{\partial\left( {I - {K_{t}F}} \right)}{\partial k_{t}}{S\left( {{K_{b}F} - K_{a}} \right)}}} \\{= \quad {{S\frac{\partial K_{t}}{\partial k_{t}}{{FS}\left( {{K_{b}F} - K_{a}} \right)}} = {S\frac{\partial K_{t}}{\partial k_{t}}{FA}}}}\end{matrix} & \text{(31c)}\end{matrix}$

where ∂K/∂k (∂K_(b)/∂k_(b), ∂K_(a)/∂k_(a) and ∂K_(c)/∂k_(c)) will haveonly one non-zero element of unity value (the nth diagonal element).Therefore, if W^(T)=(w₁, w, . . . ) and Z^(T)=(z₁, z₂. . . ) are twoarbitrary vectors, the product of W^(T) (∂K/∂k)Z=w_(n)z_(n) will be assimple as the product of two scalars. Because of this feature, ∂λ/∂κ canbe evaluated very easily as shown below.

Define the scalar

h=V^(T)U=(v₁, v₂, . . . ) (u₁, u₂, . . . )^(T)

and the vectors

W=V^(T)S=(w₁, w₂, . . . )  (32a)

Z=FU=(z₁, z₂, . . . )^(T)  (32b)

where S and F are already available in (27) and (25), respectively, whenforming the state space equation. Substituting (31) in (15),$\begin{matrix}{\frac{\partial\lambda}{\partial k_{b}} = {\frac{V^{T}S\frac{\partial K_{b}}{\partial k_{b}}{FU}}{V^{T}U} = {\frac{W^{T}\frac{\partial K_{b}}{\partial k_{b}}Z}{V^{T}U} = \frac{w_{n}z_{n}}{h}}}} & \text{(33a)} \\{{\frac{\partial\lambda}{\partial k_{a}} = {\frac{{- V^{T}}S\frac{\partial K_{a}}{\partial k_{a}}U}{V^{T}U} = {- \frac{w_{n}u_{n}}{h}}}}{or}} & \text{(33b)} \\{{\frac{\partial\lambda}{\partial k_{t}} = {\frac{V^{T}S\frac{\partial K_{t}}{\partial k_{t}}{FAU}}{V^{T}U} = {\frac{V^{T}S\frac{\partial K_{t}}{\partial k_{t}}{FU}\quad \lambda}{V^{T}U} = \frac{w_{n}z_{n}\lambda}{h}}}}{{{{Hence}\quad {for}\quad \kappa} = b},a,T_{b},{{or}\quad T_{a}},{{{\partial\lambda}/{\partial\kappa}}\quad {becomes}},{{using}(22)},}} & \text{(33c)} \\{\frac{\partial\lambda}{\partial b} = {{\frac{\partial\lambda}{\partial k_{b}}\frac{\partial k_{b}}{\partial b}} = {{\frac{\partial\lambda}{\partial k_{b}}\frac{1}{T_{a}}} = \frac{w_{n}z_{n}}{{hT}_{a}}}}} & \text{(34a)} \\{\frac{\partial\lambda}{\partial a} = \frac{{- w_{n}}u_{n}}{{hT}_{a}}} & \text{(34b)} \\{{\frac{\partial\lambda}{\partial T_{b}} = \frac{w_{n}z_{n}\lambda}{{hT}_{a}}}{or}} & \text{(34c)} \\\begin{matrix}{\frac{\partial\lambda}{\partial T_{a}} = \quad {{\frac{\partial\lambda}{\partial k_{b}}\frac{\partial k_{b}}{\partial T_{a}}} + {\frac{\partial\lambda}{\partial k_{a}}\frac{\partial k_{a}}{\partial T_{a}}} + {\frac{\partial\lambda}{\partial k_{t}}\frac{\partial k_{t}}{\partial T_{a}}}}} \\{= \quad {{- \left( {{\frac{\partial\lambda}{\partial k_{b}}b} + {\frac{\partial\lambda}{\partial k_{a}}a} + {\frac{\partial\lambda}{\partial k_{t}}T_{b}}} \right)}/T_{a}^{2}}} \\{= \quad {{- {w_{n}\left( {{z_{n}b} - {u_{n}a} + {\lambda \quad z_{n}T_{b}}} \right)}}/\left( {hT}_{a}^{2} \right)}}\end{matrix} & \text{(34d)}\end{matrix}$

Consequently, the sensitivities of λ, with respect to the nth block, areobtained by picking up nth elements from the vectors U, W arid Z andthen performing a simple algebraic operation with some other scalarsaccording to (34), where W and Z can be derived from V and U in (32).Therefore the sensitivity with respect to all block parameters can beevaluated once theses vectors are formed.

9.6.2 Sensitivities for Arbitrary System Parameters

With diagonal matrix K of (19) collecting all the parameters of zeroorder blocks, parameters associated with system operating condition areall included in K. From (25), (26), and (27), ∂A/∂κ can be expressed as$\begin{matrix}{\frac{\partial A}{\partial\kappa} = {\frac{\partial\left\lbrack {S\left( {{K_{b}F} - K_{a}} \right)} \right\rbrack}{\partial\kappa} = {S\left( {{K_{t}\frac{\partial F}{\partial\kappa}A} + {K_{b}\frac{\partial F}{\partial\kappa}}} \right)}}} & \text{(35a)} \\{\frac{\partial F}{\partial\kappa} = {\frac{\partial\left( {L_{1} + {L_{3}H\quad L_{7}}} \right)}{\partial\kappa} = {L_{3}H\frac{\partial K}{\partial\kappa}\left( {{L_{9}^{\prime}H} + I} \right)L_{7}^{\prime}}}} & \text{(35b)}\end{matrix}$

Investigating the elements of K associated with system operatingcondition, some of them are the simple functions of variables associatedwith generator buses, which are collected in M₁ of (16) or M_(1(i)) of(36) corresponding to i-th machine.

M _(1(i)) =[I _(di) , I _(qi) , V _(di) , V _(qi) , V _(ti), cos δ_(i),sin δ_(i)]  (36)

The others describing network equation are collected in M₂ of (16). With∂K/∂κ constructed from the partial derivatives of elements of M₁ and M₂,∂λ/∂κ for κ=(system parameter) therefore can be evaluated from (15) and(35).

9.6.2.1 Derivatives of M_(1(i)) to j-th Nodal Voltages (V_(Rj)+jV_(Jj))

Firstly, assuming that the nodal voltages and generator currents aredefined in rectangular coordinates. E_(Qi) can also be expressedreferencing network frame as: $\begin{matrix}{{{E_{QRi} + {j\quad E_{QJi}}} = {\left( {V_{Ri} + {j\quad V_{Ji}}} \right) + {\left( {I_{Ri} + {j\quad I_{Ji}}} \right)\left( {R_{ai} + {j\quad X_{qi}}} \right)}}},{E_{QRi} = {V_{Ri} + {I_{Ri}R_{ai}} - {I_{Ji}X_{qi}}}},{E_{QJi} = {V_{Ji} + {I_{Ri}X_{qi}} + {I_{Ji}R_{ai}}}}} & (37) \\{{E_{Qi}^{2} = {E_{QRi}^{2} + E_{QJi}^{2}}},{E_{QRi} = {{- E_{Qi}}\quad \sin \quad \delta_{i}}},{E_{QJi} = {E_{Qi}\quad \cos \quad \delta_{i}}}} & (38)\end{matrix}$

From the nodal voltage equation of overall network I=YV, coordinatestransformation (11) and, $\begin{matrix}{V_{ri}^{2} = {V_{Ri}^{2} + V_{Ji}^{2}}} & (39)\end{matrix}$

the partial derivatives can be expressed as follows by using a constantC_(o) with C_(o)=1 for i=j and C_(o)=o for i≠j. Subscript “(.)j” standsfor “Rj” or “Jj”. $\begin{matrix}{\frac{\partial I_{Ri}}{\partial V_{Rj}} = {\frac{\partial I_{Ji}}{\partial V_{Jj}} = G_{ij}}} & \text{(40a)} \\{\frac{\partial I_{Ri}}{\partial V_{Jj}} = {{- \frac{\partial I_{Ji}}{\partial V_{Rj}}} = {- B_{ij}}}} & \text{(40b)} \\{\frac{\partial E_{QRi}}{\partial V_{Rj}} = {\frac{\partial E_{QJi}}{\partial V_{Jj}} = {C_{0} + {R_{ai}\frac{\partial I_{Ri}}{\partial V_{Rj}}} - {X_{qi}\frac{\partial I_{Ji}}{\partial V_{Rj}}}}}} & \text{(41a)} \\{\frac{\partial E_{QRi}}{\partial V_{Jj}} = {{- \frac{\partial E_{QJi}}{\partial V_{Rj}}} = {{R_{ai}\frac{\partial I_{Ri}}{\partial V_{Jj}}} - {X_{qi}\frac{\partial I_{Ji}}{\partial V_{Jj}}}}}} & \text{(41b)} \\{{\frac{\partial E_{Qi}}{\partial V_{{(.)}j}} = {{{- \frac{\partial E_{QRi}}{\partial V_{{(.)}j}}}\sin \quad \delta_{i}} + {\frac{\partial E_{QJi}}{\partial V_{{(.)}j}}\cos \quad \delta_{i}}}}{{\frac{{\partial\sin}\quad \delta_{i}}{\partial V_{{(.)}j}} = {\frac{- 1}{E_{Qi}}\left( {\frac{\partial E_{QRi}}{\partial V_{{(.)}j}} + {\frac{\partial E_{Qi}}{\partial V_{{(.)}j}}\sin \quad \delta_{i}}} \right)}},}} & \text{(41c)} \\{{\frac{{\partial\cos}\quad \delta_{i}}{\partial V_{{(.)}j}} = {\frac{1}{E_{Qi}}\left( {\frac{\partial E_{QJi}}{\partial V_{{(.)}j}} - {\frac{\partial E_{Qi}}{\partial V_{{(.)}j}}\cos \quad \delta_{i}}} \right)}}\begin{matrix}{\frac{\partial I_{di}}{\partial V_{{(.)}j}} = \quad {{\frac{\partial I_{Ri}}{\partial V_{{(.)}j}}\cos \quad \delta_{i}} + {\frac{\partial I_{Ji}}{\partial V_{{(.)}j}}\sin \quad \delta_{i}} +}} \\{\quad {{{I_{Ri}{\partial\cos}\quad \frac{\delta_{i}}{\partial V_{{(.)}j}}} + {I_{Ji}{\partial\sin}\quad \frac{\delta_{i}}{\partial V_{{(.)}j}}}},}}\end{matrix}} & (42) \\{\begin{matrix}{\frac{\partial I_{qi}}{\partial V_{{(.)}j}} = \quad {{{- \frac{\partial I_{Ri}}{\partial V_{{(.)}j}}}\sin \quad \delta_{i}} + {\frac{\partial I_{Ji}}{\partial V_{{(.)}j}}\cos \quad \delta_{i}} -}} \\{\quad {{I_{Ri}{\partial\sin}\quad \frac{\delta_{i}}{\partial V_{{(.)}j}}} + {I_{Ji}{\partial\cos}\quad \frac{\delta_{i}}{\partial V_{{(.)}j}}}}}\end{matrix}{{\frac{\partial V_{di}}{\partial V_{Rj}} = {{C_{0}\cos \quad \delta_{i}} + {V_{Ri}{\partial\cos}\quad \frac{\delta_{i}}{\partial V_{Rj}}} + {V_{Ji}{\partial\sin}\quad \frac{\delta_{i}}{\partial V_{Rj}}}}},{\frac{\partial V_{di}}{\partial V_{Jj}} = {{C_{0}\sin \quad \delta_{i}} + {V_{Ri}{\partial\cos}\quad \frac{\delta_{i}}{\partial V_{Jj}}} + {V_{Ji}{\partial\sin}\quad \frac{\delta_{i}}{\partial V_{Jj}}}}},{\frac{\partial V_{qi}}{\partial V_{Rj}} = {{{- C_{0}}\sin \quad \delta_{i}} - {V_{Ri}{\partial\sin}\quad \frac{\delta_{i}}{\partial V_{Rj}}} + {V_{Ji}{\partial\cos}\quad \frac{\delta_{i}}{\partial V_{Rj}}}}},}} & (43) \\{\frac{\partial V_{qi}}{\partial V_{Jj}} = {{C_{0}\cos \quad \delta_{i}} - {V_{Ri}{\partial\sin}\quad \frac{\delta_{i}}{\partial V_{Jj}}} + {V_{Ji}{\partial\cos}\quad \frac{\delta_{i}}{\partial V_{Jj}}}}} & (44) \\{{\frac{\partial V_{ti}}{\partial V_{Rj}} = {C_{0}\frac{V_{Ri}}{V_{ti}}}},{\frac{\partial V_{ti}}{\partial V_{Jj}} = {C_{0}\frac{V_{Ji}}{V_{ti}}}}} & (45)\end{matrix}$

9.6.2.2 Derivatives of M₂ to Nodal Voltages (V_(Rj)+jV_(Jj))

M₂ of (16) will be consisted of the elements of Y_(GG)′ alone when themachine is considered only. When static loads are represented byequivalent shunt admittances and added to the corresponding diagonalelements of Y_(LL)′ of (4), only the equivalent admittances are affectedby nodal voltages. Thus, $\begin{matrix}\begin{matrix}{\frac{\partial Y_{GG}^{\prime}}{\partial V_{{(.)}j}} = \quad {\frac{\partial}{\partial V_{{(.)}j}}\left\lbrack {Y_{GG} - {{Y_{GL}\left( Y_{LL}^{\prime} \right)}^{- 1}Y_{LG}}} \right\rbrack}} \\{= \quad {Y_{GL}Y_{LL}^{- 1}\frac{\partial Y_{LL}^{\prime}}{\partial V_{{(.)}j}}Y_{LL}^{- 1}Y_{LG}}}\end{matrix} & (46)\end{matrix}$

When the deviation of nodal voltages is regarded being independent withload powers, differentiation of (47a) yields (47b). $\begin{matrix}{{{\Delta \quad G_{ii}} + {j\quad \Delta \quad B_{ii}}} = {\left( {P_{l,i} - {j\quad Q_{l,i}}} \right)/V_{ti}^{2}}} & \text{(47a)} \\{\frac{\partial\left( {{\Delta \quad G_{ii}} + {{j\Delta}\quad B_{ii}}} \right)}{\partial V_{{(.)}i}} = {\frac{{- 2}V_{{(.)}i}}{V_{ti}^{4}}\left( {P_{l,i} - {j\quad Q_{l,i}}} \right)}} & \text{(47b)}\end{matrix}$

If the deviation of nodal voltages is risen from the variation of nodalpowers, the equivalent admittances should be expressed by nodal injectedcurrents as: $\begin{matrix}{{{\Delta \quad G_{ii}} + {j\quad \Delta \quad B_{ii}}} = {- \frac{I_{Ri} + {j\quad I_{Ji}}}{V_{Ri} + {j\quad V_{Ji}}}}} & (48)\end{matrix}$

and the injected currents are determined by nodal voltage equation I=YV.Therefore, $\begin{matrix}{{\frac{{\partial\Delta}\quad G_{ii}}{\partial V_{{(.)}j}} = {\frac{- 1}{V_{ti}^{2}}\left\lbrack {{C_{0}\left( {2V_{{(.)}j}\Delta \quad {G_{ii} \div I_{{(.)}j}}} \right)} + {V_{Ri}{\frac{\partial I_{Ri}}{\partial V_{{(.)}j}} \div V_{Ji}}\frac{\partial I_{Ji}}{\partial V_{{(.)}j}}}} \right\rbrack}},{\frac{{\partial\Delta}\quad B_{ii}}{\partial V_{Rj}} = {\frac{- 1}{V_{ti}^{2}}\left\lbrack {{{{C_{0}\left( {2V_{Ri}\Delta \quad {B_{ii} \div I_{Ji}}} \right)} \div V_{Ri}}\frac{\partial I_{Ji}}{\partial V_{Rj}}} - {V_{Ji}\frac{\partial I_{Ri}}{\partial V_{Rj}}}} \right\rbrack}},{\frac{{\partial\Delta}\quad B_{ii}}{\partial V_{Jj}} = {\frac{- 1}{V_{ti}^{2}}\left\lbrack {{C_{0}\left( {{2V_{Ji}\Delta \quad B_{ii}} - I_{Ri}} \right)} + {V_{Ri}\frac{\partial I_{Ji}}{\partial V_{Jj}}} - {V_{Ji}\frac{\partial I_{Ri}}{\partial V_{Jj}}}} \right\rbrack}}} & (49)\end{matrix}$

9.6.2.3 Derivatives to Nodal Voltages (V_(j), δ_(vj))

When nodal voltages in polar coordinates are defined as V_(j)∠δ_(vj),the derivatives of any element M_(i) in and M₁ or M₂ of (16) can beobtained from V_(Rj)=V_(j) cosδ_(vj) and V_(Jj)=V_(j) cosδ_(vJ) as:$\begin{matrix}{{\frac{\partial M_{i}}{\partial V_{j}} = {{\frac{\partial M_{i}}{\partial V_{Rj}}\cos \quad \delta_{vj}} + {\frac{\partial M_{i}}{\partial V_{Jj}}\sin \quad \delta_{vj}}}},{\frac{\partial M_{i}}{\partial\delta_{vj}} = {{{- \frac{\partial M_{i}}{\partial V_{Rj}}}V_{j}\sin \quad \delta_{vj}} + {\frac{\partial M_{i}}{\partial V_{Jj}}V_{j}\cos \quad {\delta_{vj}.}}}}} & (50)\end{matrix}$

9.6.2.4 Derivatives of K to Nodal Injected Power (P_(j)+Q_(j))

Jacobian matrix J used in load flow calculation determines thelinearized relationship between nodal powers and nodal voltages.∂V_(Ri)/∂P_(j), ∂V_(Ri)/∂Q_(j), ∂V_(Ji)/∂P_(j) and ∂V_(Ji)/∂Q_(j) arethe elements of J⁻¹. Therefore, the derivatives of any element M_(i) inand M₁ or M₂ of (16) to nodal powers will be expressed as:$\begin{matrix}{{\frac{\partial M_{i}}{\partial P_{j}} = {\sum\limits_{i}\left( {{\frac{\partial M_{i}}{\partial V_{Ri}}\frac{\partial V_{Ri}}{\partial P_{j}}} + {\frac{\partial M_{i}}{\partial V_{Ji}}\frac{\partial V_{Jj}}{\partial P_{j}}}} \right)}}{\frac{\partial M_{i}}{\partial Q_{j}} = {\sum\limits_{i}\left( {{\frac{\partial M_{i}}{\partial V_{Ri}}\frac{\partial V_{Ri}}{\partial Q_{j}}} + {\frac{\partial M_{i}}{\partial V_{Ji}}\frac{\partial V_{Jj}}{\partial Q_{j}}}} \right)}}} & (51)\end{matrix}$

9.6.2.5 Derivatives of K to Line Admittance (y_(mn)=g_(mn)+jb_(mn))

Change of a line admittance y_(mn) will affect admittance matrix Y andthe initial system operating condition described by nodal voltages andinjected currents. Derivative of K with respect to b_(mn) has the sameform as that to g_(mn). Only the latter is discussed.

The derivatives of Y_(GG)′ to g_(mn) will be obtained from ∂Y/∂g_(mn)which is constructed as: “1” on the sits of (m,m) and (n,n), “−1” on thesits of (m,n) and (n,m), “0” on other sites The derivative of Y_(GG)′with respect to g_(mn) is: $\begin{matrix}{\frac{\partial Y_{GG}^{\prime}}{\partial g_{mn}} = {\frac{\partial Y_{GG}}{\partial g_{mn}} - {\frac{\partial Y_{GL}}{\partial g_{mn}}\left( Y_{LL}^{\prime} \right)^{- 1}Y_{LG}} + {{Y_{GL}\left( Y_{LL}^{\prime} \right)}^{- 1}\frac{\partial Y_{LL}^{\prime}}{\partial g_{mn}}\left( Y_{LL}^{\prime} \right)^{- 1}Y_{LG}} - {{Y_{GL}\left( Y_{LL}^{\prime} \right)}^{- 1}\frac{\partial Y_{LG}}{\partial g_{mn}}}}} & (52)\end{matrix}$

However, due to I=YV, (40) will become (in vector form) $\begin{matrix}{\frac{\partial I}{\partial g_{\min}} = {{Y\frac{\partial V}{\partial g_{mn}}} + {\frac{\partial Y}{\partial g_{mn}}V}}} & (53)\end{matrix}$

From (41) to (45), if ∂M_(i)/∂V is regarded as a linear function of∂I/∂V denoted as f(.), the derivatives of M_(i) will be represented invector form as $\begin{matrix}{\frac{\partial M_{i}}{\partial g_{mn}} = {{{f\left( \frac{\partial I}{\partial V} \right)}\frac{\partial V}{\partial g_{mn}}} + {f\left( {\frac{\partial Y}{\partial g_{mn}}V} \right)}}} & (54)\end{matrix}$

∂V/∂g_(mn) can be solved from (55) which is obtained by differentiatingthe nodal power equation. $\begin{matrix}{{{{\frac{\partial V_{i}}{\partial g_{mn}}I_{i}^{*}} + {\sum\limits_{j}\left( {{V_{i}\frac{\partial Y_{ij}^{*}}{\partial g_{mn}}V_{j}^{*}} + {V_{i}Y_{ij}^{*}\frac{\partial V_{j}^{*}}{\partial g_{mn}}}} \right)}} = 0},{i = 1},2,3,\ldots} & (55)\end{matrix}$

9.7 Single 3rd-Order Machine Infinite Busbar Model

The single 3-rd order machine infinite busbar model can be establishedby connecting the 3rd-order machine module in FIG. 4 with the tielineinfinite busbar network module (FIG. 13) [25]. The reference frame isstationary, that is Δ, Ω_(ref)=0.

We claim:
 1. A model for testing an entire power transmission systemunder small perturbation stability comprising: a microprocessor forconducting stability analyses of said power system; a plurality of firsttype of plug-in transfer blocks representing various components of saidpower system and processable by said microprocessor wherein each blockis derived from sets of equations representing one of said components; arecombination network represented by a second type of transfer blocks,wherein said second type of block is derived from sets of networkequations representing the power system network and said second type ofblock is adapted to receive a first set of said first type of plug-inblocks to form a model of a power system, such that an entire powersystem may be tested for small perturbation stability by simple matrixmanipulation when processed by said microprocessor.
 2. A method formodeling a power transmission system under small perturbation stabilitycomprising the steps of: a. providing a microprocessor for conductingstability analyses of said power system; b. preparing a plurality offirst type of plug-in transfer blocks representing various components ofsaid power system and processable by said microprocessor wherein eachblock is derived from sets of equations representing one of saidcomponents; c. providing a recombination network represented by a secondtype of transfer blocks, wherein said second type of block is derivedfrom sets of network equations representing the power system network andsaid second type of block is adapted to receive a first set of saidfirst type of plug-in blocks to form a model of a power system, suchthat an entire power system may be tested for small perturbationstability by simple matrix manipulation d. matrix manipulating of saidrecombination network to test for small perturbation stability of anentire power system.